Nearest just interval: Difference between revisions
Wikispaces>genewardsmith **Imported revision 177732917 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 177738501 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-09 00:30:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>177738501</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger. | ||
Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest. | Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest. | ||
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==Examples== | ==Examples== | ||
The 600-cent interval sqrt(2) (6 steps of [[12edo]], "Tritone") approximates following ratios: | The 600-cent interval sqrt(2) (6 steps of [[12edo]], "Tritone") approximates following ratios: | ||
|| **freq. ratio** || ** | || **freq. ratio** || **log2([[Tenney Height]])** || **size** in cents || **"error"** in cents || | ||
|| ... || ... || ... || ... || | || ... || ... || ... || ... || | ||
||= 1 / 1 ||= 0.0 ||= 0.0 ||= 600.0|| | |||
||= 3 / 2 ||= 2.585 ||= 701.96 ||= 101.96 || | ||= 3 / 2 ||= 2.585 ||= 701.96 ||= 101.96 || | ||
||= 4 / 3 ||= 3.585 ||= 498.04 ||= 101.96 || | |||
||= 7 / 5 ||= 5.129 ||= 582.51 ||= 17.49 || | ||= 7 / 5 ||= 5.129 ||= 582.51 ||= 17.49 || | ||
||= 17 / 12 ||= 7.672 ||= 603.00 ||= 3.000 || | ||= 17 / 12 ||= 7.672 ||= 603.00 ||= 3.000 || | ||
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|| **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents || | || **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents || | ||
|| ... || ... || ... || ... || | || ... || ... || ... || ... || | ||
||= 1 / 1 ||= 0.0 ||= 0.0 ||= 300.0|| | |||
||= 6 / 5 ||= 4.907 ||= 315.64 ||= 15.64 || | ||= 6 / 5 ||= 4.907 ||= 315.64 ||= 15.64 || | ||
||= 13 / 11 ||= 7.160 ||= 289.21 ||= 10.79 || | |||
||= 19 / 16 ||= 8.248 ||= 297.51 ||= 2.49 || | ||= 19 / 16 ||= 8.248 ||= 297.51 ||= 2.49 || | ||
||= 25 / 21 ||= 9.036 ||= 301.84 ||= 1.84 || | ||= 25 / 21 ||= 9.036 ||= 301.84 ||= 1.84 || | ||
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</pre></div> | </pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. <br /> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.<br /> | ||
<br /> | <br /> | ||
Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.<br /> | Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number of interest such as 3/2 or 5^(1/4) is often of interest.<br /> | ||
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<td><strong>freq. ratio</strong><br /> | <td><strong>freq. ratio</strong><br /> | ||
</td> | </td> | ||
<td><strong> | <td><strong>log2(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br /> | ||
</td> | </td> | ||
<td><strong>size</strong> in cents<br /> | <td><strong>size</strong> in cents<br /> | ||
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</td> | </td> | ||
<td>...<br /> | <td>...<br /> | ||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1 / 1<br /> | |||
</td> | |||
<td style="text-align: center;">0.0<br /> | |||
</td> | |||
<td style="text-align: center;">0.0<br /> | |||
</td> | |||
<td style="text-align: center;">600.0<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">701.96<br /> | <td style="text-align: center;">701.96<br /> | ||
</td> | |||
<td style="text-align: center;">101.96<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">4 / 3<br /> | |||
</td> | |||
<td style="text-align: center;">3.585<br /> | |||
</td> | |||
<td style="text-align: center;">498.04<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">101.96<br /> | <td style="text-align: center;">101.96<br /> | ||
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</td> | </td> | ||
<td>...<br /> | <td>...<br /> | ||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1 / 1<br /> | |||
</td> | |||
<td style="text-align: center;">0.0<br /> | |||
</td> | |||
<td style="text-align: center;">0.0<br /> | |||
</td> | |||
<td style="text-align: center;">300.0<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">15.64<br /> | <td style="text-align: center;">15.64<br /> | ||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">13 / 11<br /> | |||
</td> | |||
<td style="text-align: center;">7.160<br /> | |||
</td> | |||
<td style="text-align: center;">289.21<br /> | |||
</td> | |||
<td style="text-align: center;">10.79<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||